Problem 48 - Prealgebra

Question

In Brady's math class, approximately \(\frac{3}{5}\) of the students have pets. About 41 out of every 50 students in his school have pets. Do a greater fraction of students have pets in Brady's math class or in his school?

Step-by-Step Solution

Answer

A greater fraction of students have pets in the school (
{41}{50}).

Step 1: Compare Fractions

We are given fractions for two groups: Brady's math class \( \frac{3}{5} \) and his school where 41 out of 50 students have pets, which can be written as the fraction \( \frac{41}{50} \). To compare these fractions, we'll find a common denominator.

Step 2: Find Common Denominator

The denominators are 5 and 50. The least common multiple of 5 and 50 is 50. So, we need to convert \( \frac{3}{5} \) to an equivalent fraction with a denominator of 50.

Step 3: Convert \\(\frac{3}{5}\\) to an Equivalent Fraction with Denominator 50

Multiply both the numerator and the denominator of \(\frac{3}{5}\) by 10 to get the equivalent fraction \(\frac{30}{50}\). Now we have \(\frac{30}{50}\) for the math class and \(\frac{41}{50}\) for the school.

Step 4: Compare the Equivalent Fractions

Now that both fractions have the same denominator, we can compare the numerators directly. We compare 30 (Brady's math class) and 41 (school). Since 30 < 41, \(\frac{30}{50} \) is less than \(\frac{41}{50} \).

Step 5: Conclusion

Since the fraction of students with pets in the school \(\frac{41}{50}\) is greater than that in Brady's math class \(\frac{30}{50}\), a greater fraction of students have pets in his school.

Key Concepts

Equivalent Fractions

Equivalent fractions are different fractions that represent the same value or proportion of a whole. This concept is crucial when comparing fractions, especially when they have different denominators. To create an equivalent fraction, you can multiply or divide both the numerator and the denominator by the same non-zero number.

• For example, if you have \(\frac{3}{5}\), you can find an equivalent fraction by multiplying the numerator and the denominator by the same number, such as 10. This gives you \(\frac{30}{50}\).
• Equivalent fractions allow you to compare fractions easily, especially when finding a common denominator.
• It's important to remember that although the numbers may change, the value of the fraction does not.

Understanding equivalent fractions is fundamental for adding, subtracting, and especially comparing fractions with different denominators.

Least Common Multiple

The least common multiple (LCM) is the smallest positive integer that is divisible by each of the numbers in question. Finding the LCM is a helpful step when comparing fractions because it helps determine a common denominator for fractions with different denominators.

• To find the LCM of two numbers, such as 5 and 50, list the multiples of each number until you find the smallest common multiple.
• For 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50...
• For 50: 50, 100, 150...
• The smallest common multiple is 50.

By using the least common multiple when comparing fractions, you ensure that both fractions have the same denominator, which makes comparing their numerators straightforward and accurate.

Fraction Comparison

Fraction comparison involves determining which of two or more fractions represents a larger portion of a whole. The best method to compare fractions is to find a common denominator, allowing for direct comparison of numerators.

• For instance, to compare the fractions \(\frac{3}{5}\) and \(\frac{41}{50}\), convert \(\frac{3}{5}\) into an equivalent fraction with a denominator of 50.
• Multiply the numerator and denominator of \(\frac{3}{5}\) by 10, providing \(\frac{30}{50}\).
• Now, compare \(\frac{30}{50}\) and \(\frac{41}{50}\).
• The fraction \(\frac{41}{50}\) is larger because 41 is greater than 30.

Comparing fractions is a valuable skill for understanding proportions and making informed judgments about sizes and amounts.